The use of spinors makes it possible to formulate a spin \(1/2\)  Hamiltonian since spinors transform in the fundamental representation of \(\mathrm {SU}(2)\) . Such a Hamiltonian is expected (and was shown in [1]}) to generate a recursion relation of the \(\lbrace 6j\rbrace \) -symbols with arguments changed by \(1/2\) . As proposed in [1]}, the spin \(1/2\)  Hamiltonian (denoted with a superscript \((1/2)\) ) represents the unchanged inner product of the spinors \(\tilde{z}_2\)  and \(\tilde{z}_6\)  after transported around the plaquette surrounded by links \(e_1,e_2,e_6\) ,
\(H^{(1/2)}_{126}:=\langle \tilde{z}_6|\tilde{z}_2] [\tilde{z}_6 | g_6g_1g_2^{-1} -\mathbb {I}|\tilde{z}_2\rangle \,.\) 
